Question

gk ≅ hk and jk ≅ jk. complete the proof that ∠j ≅ ∠i.
statement
1 gk ≅ hk
2 jk ≅ jk
3 ∠gkj ≅ ∠hki
4 △gjk ≅ △hik

Explanation:

Step1: Given

Given that \(\overline{GK}\cong\overline{HK}\) and \(\overline{JK}\cong\overline{IK}\) (from the problem statement).

Step2: Vertical angles

\(\angle GKJ\) and \(\angle HKI\) are vertical angles. By the vertical - angles theorem, \(\angle GKJ\cong\angle HKI\).

Step3: SAS (Side - Angle - Side) congruence

In \(\triangle GJK\) and \(\triangle HIK\), we have \(\overline{GK}\cong\overline{HK}\), \(\angle GKJ\cong\angle HKI\), and \(\overline{JK}\cong\overline{IK}\). So, by the SAS (Side - Angle - Side) congruence criterion, \(\triangle GJK\cong\triangle HIK\).

Step4: Corresponding parts of congruent triangles

Since \(\triangle GJK\cong\triangle HIK\), by the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, \(\angle J\cong\angle I\).

Answer:

\(\angle J\cong\angle I\) (by CPCTC as \(\triangle GJK\cong\triangle HIK\) via SAS)